ON THE AVERAGE ECCENTRICITY, THE HARMONIC INDEX an‎d THE LARGEST SIGNLESS LAPLACIAN EIGENVALUE OF A GRAPH

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is dened as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of n-vertex trees with two adjacent vertices of maximum degree Δ, where n 2Δ. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randic index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas .